Risk Analysis of Industrial Structures Under Extreme Transient Loads
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Transcript of Risk Analysis of Industrial Structures Under Extreme Transient Loads
Risk analysis of industrial structures under extreme transient loads
D.G. Talaslidisa,*, G.D. Manolisa, E. Paraskevopoulosa, C. Panagiotopoulosa,N. Pelekasisb, J.A. Tsamopoulosc
aLaboratory of Applied Statics, Department of Civil Engineering, Aristotle University, Thessaloniki GR-54124, GreecebLaboratory of Fluid Mechanics, Department of Mechanical Engineering, University of Thessaly, Volos GR-38334, Greece
cLaboratory of Computational Fluid Dynamics, Department of Chemical Engineering, University of Patras, Patras GR-26500, Greece
Accepted 20 February 2004
Abstract
A modular analysis package is assembled for assessing risk in typical industrial structural units such as steel storage tanks, due to extreme
transient loads that are produced either as a result of chemical explosions in the form of atmospheric blasts or because of seismic activity in
the form of ground motions. The main components of the methodology developed for this purpose are as follows: (i) description of blast
overpressure and ground seismicity, (ii) transient non-linear finite element analysis of the industrial structure, (iii) development of 3D-
equivalent, continuous beam multi-degree-of-freedom structural models, (iv) introduction of soil–structure interaction effects,
(v) probabilistic description of the loading process and the stiffness/mass characteristics of the structure, and (vi) generation of fragility
curves for estimation of structural damage levels by using the Latin hypercube statistical sampling method. These fragility curves can then be
used within the context of the engineering analysis–design cycle, so as to minimize structural failure probability under both man-made
hazards such as blasts and natural hazards such as earthquake-induced transient loads.
q 2004 Elsevier Ltd. All rights reserved.
Keywords: Chemical explosions; Industrial structures; Finite elements; Fragility curves; Latin hypercube method; Risk analysis; Seismic loads; Structural
dynamics
1. Introduction
The development of methods for risk analysis based on
probabilistic concepts in the presence of hazards such as
seismically induced ground motions or chemically induced
atmospheric blasts is a prerequisite for damage assessment
in key structures (storage tanks, piping systems, industrial
buildings, loading docks, etc.) that comprise large industrial
complexes [1–8]. Current design codes and guidelines
cannot completely cover such issues because much
uncertainty is involved, primarily with respect to the nature
of the various accident scenarios that are possible [9,10].
Further investigations are, therefore, required in order to
better understand the risk involved and to obtain results that
can be used in formulating comprehensive intervention
strategies. Since the aforementioned structural units form
an integral part of larger installations, improvement in
safety helps to ensure a reliable and continuous operation of
the entire industrial complex, which in turn implies better
environmental protection [11].
Current methods of structural analysis that assume a
deterministic environment selectively utilize part of all
available information regarding loading, material
parameters and structural configuration and are thus capable
of producing only representative (e.g. maximum or
minimum) values for the structural response [3].
The introduction of probabilistic models, especially in the
description of the structure itself, requires a different level
of analysis in order to provide quantitative information on
structural reliability and on risk of failure. Furthermore, it
is essential that any increase in sophistication implied for
the loading is not compromised by simplification in the
description of the structural model, i.e. the level of accuracy
in all steps of the analysis must be adjusted so as to produce
an all-around consistent mechanical model [2].
Recent work in this area includes use of extensive Monte
Carlo simulations (MCS) for the seismic finite element
0267-7261/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.soildyn.2004.02.003
Soil Dynamics and Earthquake Engineering 24 (2004) 435–448
www.elsevier.com/locate/soildyn
* Corresponding author. Tel.: þ30-2310-99-5707; fax: 30-2310-99-
5769.
E-mail addresses: [email protected] (D.G. Talaslidis), tsamos@
chemeng.upatras.gr (J.A. Tsamopoulos).
method (FEM) analysis of nuclear reactor facilities,
including soil–structure interaction (SSI) effects [12].
Uncertainty is assumed to exist in the free-field motion,
the local site conditions and the structural parameters, which
are modelled by uniform, Gaussian and log-normal
distributions. These random processes are in turn
represented by their respective Karhunen–Loeve expan-
sions and results are cast in terms of probabilistic
in-structure spectra. Next, the non-linear stochastic dynamic
analysis of an SSI system consisting of an oscillator plus
foundation resting on a 2D soil deposit was performed using
the FEM with MCS in order to determine risk of damage
due to liquefaction under earthquakes [13]. Soil properties
were measured from cone penetration tests from a real site
and were found to match a beta-distribution. The seismic
input was modelled as a non-stationary random process and
the analysis results were presented in terms of fragility
curves. Fragility curves have also been used in conjunction
with bridges, and their computation is based on either
standard time history analyses or on simplified methods
such as the capacity spectrum approach, which is essentially
a static non-linear procedure. Specifically, 10 nominally
identical, but statistically different highway bridges were
analysed under 80 records of ground motion histories for
this purpose [14]. Also, the seismic behaviour of on-grade,
steel liquid storage cylindrical tanks subjected to
ground shaking hazard was examined and fragility curves
were developed by analysing the reported performance
of over 400 such tanks under nine separate earthquake
events [15].
As far as explosions are concerned, we mention the
construction of pressure– impulse diagrams based on
single degree-of-freedom (SDOF) representations of struc-
tural systems, where the blast load is a simplified
descending pressure pulse [16]. These diagrams are used
in identifying damage regimes in the SDOF, much like a
response spectrum and are defined in terms of the ratio
between loading time and response period of the structure.
Finally, more refined statistical sampling techniques such
as the Latin hypercube method [17] have been recently
introduced for generating variations of a typical structural
configuration for stochastic FEM analysis purposes.
These sampling procedures strongly improve the statistical
representation of stochastic design parameters as compared
to standard MCS and also help reduce spurious correlations
in the processed data.
In this work, an integrated methodology is developed for
assessing risk to a typical industrial structure with respect to
accident scenarios involving either a chemical explosion or
an earthquake. The methodology first focuses on generation
and subsequent propagation of overpressure due to chemical
explosions [18]. In parallel, we give a description of the
seismically induced ground motions [19]. As far as
the structural model is concerned, the starting point is
non-linear, transient FEM analysis of cylindrical, thin-
walled shell structures. This implies use of efficient
triangular and quadrilateral shell finite elements based on
the generalized Hu-Washizu principle within the context
of elastoplastic material response [20]. Based on the results
of the 3D finite element analyses, a simplified model in the
form of a multi-degree-of-freedom (MDOF) continuous
beam with variable thickness is developed for capturing all
salient aspects of the tank’s response. SSI effects are
accounted for by introducing simplified, yet realistic soil
impedance coefficients [21]. Next, this MDOF system with
material, geometric and loading parameters that are
stochastic variables is used for assessing risk in the presence
of random loads [22,23]. In particular, the basic failure
mode considered is material yielding with local buckling
(e.g. ‘elephant foot’) ignored. Furthermore, two basic limit
states are distinguished, namely serviceability and ultimate
strength. Earlier work along these lines [24,25] employed
MCS with SDOF structural models, which have been
replaced here by the more efficient Latin hypercube method
[17] and by a continuous beam model, respectively. The final
results of the risk analysis is generation of fragility curves
plotting limit state probability as a function of blast load or
peak ground acceleration (PGA), which allow for an
estimation of damage levels in the structure under
investigation.
Briefly, the material presented in this paper is organized
as follows. In Section 2, the load sustained by the industrial
structure shown in Fig. 1(a) is described for two cases,
namely chemical explosions and seismic motions.
Both overpressure duration and ground acceleration filter
parameters are treated as random variables, so that
realizations of the pressure and acceleration fields can be
computed within the framework of the Latin hypercube
method. Next, the dynamic behaviour of the structure is
examined by a 3D FEM model, which is used in turn to
define a simplified MDOF model with SSI. The key part is
Section 4, where all information is integrated for the
purpose of assessing risk, which is carried out within the
framework of the Latin hypercube method and conveniently
described by a set of fragility curves. Finally, some
conclusions are drawn regarding probability for the
structure to exhibit damage of a certain level at a given
external load intensity.
2. Load description
As previously mentioned, the methodology developed
herein is modular in form; each component is now
separately analysed, starting with the loading scenarios
shown in Figs. 2 and 3.
2.1. Chemical explosion model
The majority of data on blast effects are scaled in practice
with respect to the blast pressure output of a spherical
charge of TNT explosive [7]. Of interest in the context of
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448436
industrial plants is explosion of a cloud of flammable
vapour. Such explosions are usually the result of massive
spills of combustible hydrocarbons in the atmosphere,
followed by ignition and acceleration of the flame
propagating through the cloud so as to produce a destructive
shock wave. Depending on the speed of the flame front,
two different mechanisms are at work, namely deflagration
and detonation. Existing guidelines for estimating blast
damage are based on the TNT equivalent yield concept that,
in conjunction with the Rankine–Hugoniot relations [26]
allows for computation of all necessary quantities
(overpressure, reflected pressure, shock velocity, stagnation
pressure, etc.) with relative ease.
The basic assumptions made for evaluating air blast
loading on an industrial structure [18] are as follows: (i) an
ideal shock is produced in which the peak overpressure is
reached instantaneously; (ii) the structure is in the Mach
reflection region where the air blast front is propagating
parallel to the ground; (iii) the shock load can be decoupled
from the structural response, and (iv) the structure is treated
as a rigid body. The loading scenario develops as the air
Fig. 1. Steel storage tank: (a) geometry, (b) vertical cross-section, (c) finite element mesh, (d) equivalent MDOF model and (e) equivalent MDOF model
with SSI.
Fig. 2. Blast load: spatial variation of the load (a) across the horizontal
plane, (b) along the height and (c) temporal variation of the net
overpressure.
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448 437
pressure reaches the front face (first point impact) of the
circular cylindrical tank. We have an instantaneous rise
from ambient air pressure to overpressure pðtÞ felt in
the immediate surroundings of the structure, which quickly
increases in value because of the emergence of a reflected
pressure. This is followed by rapid decay as air flows around
the tank. The total pressure, which corresponds to pðtÞ þ
cdqðtÞ (with qðtÞ the dynamic pressure and cd the drag
coefficient) reaches the stagnation point at clearing time
t1 ¼ 3d=U; with d being the tank diameter and U the shock
velocity. The pressure’s spatial variation over the cylind-
rical tank’s circumference is simply a function of the angle
that the normal to a given point on the surface traces with
respect to an axis parallel to the direction of blast
propagation. Subsequently, the rear face is completely
engulfed and a maximum pressure registers there at blast
time t2 ¼ ðd þ 3hÞ=U; where h is the height of the tank.
Also, the average roof loading can be idealized as triangular
in time with peak pressure equal to cdqðtÞ: Finally, when this
overall pressure is averaged, i.e. integrated over the entire
structure, a net horizontal force PðtÞ acting on the structure
is produced, which is uniform along the tank’s height and in
the direction of the propagating front. For simplicity, a
linear drop in time is assumed for load PðtÞ that lasts for tdseconds, with maximum intensity P0 corresponding to 2/3
of the value obtained from integrating the total pressure on
the tank in order to account for suction in the rear surfaces.
This load model is shown in Fig. 2(c) and is a somewhat
simplified version of the one used in earlier work [24,25],
but retains all salient features necessary for a rational
analysis of the tank.
As far as the statistical properties of the blast load are
concerned, we assume that time duration td of the net
horizontal force PðtÞ is a random variable with uniform
distribution. We furthermore define six load levels, which
depend on the strength of the explosion that took place.
These load levels are calibrated to produce structural
damage that ranges from minimal to severe. It thus becomes
possible to generate a family of statistically equivalent,
time-dependent net horizontal forces for each load level,
which act uniformly across the height of the structure
(for both the 3D FEM tank model as well as the continuous
beam model derived from it), all within the context of the
Latin hypercube sampling method [17].
2.2. Seismic motion model
The simple, yet commonly used model of horizontally
polarized, elastic shear (SH) waves is adopted herein.
Soil layers over bedrock are modelled by the Kanai–Tajimi
(KT) and by the high-pass (HP) filters in the frequency v
domain. Thus, the earthquake signal at the source is viewed
as a broad-banded, white noise stochastic process with a
constant power spectral density function (PSDF) S0:
This function is subsequently filtered through soil to yield
the PSDF at the surface of the ground as
SgðvÞ ¼ HkðvÞHhðvÞS0; ð1Þ
HkðvÞ ¼ {1 þ 4z2kðv=vkÞ
2}={½1 2 ðv=vkÞ2�2 þ 4z2
kðv=vkÞ2}
and
HhðvÞ ¼ ðv=vhÞ2={½1 2 ðv=vhÞ
2�2 þ 4z2hðv=vhÞ
2}
In the above, fundamental frequency vk and damping
ratio zk for the KT filter, respectively, range from 3p to 9p
(rad/s) and from 15 to 25% so as to cover a wide range of
soil configurations. As far as the HP filter is concerned, these
values are held fixed at vh ¼ p (rad/s) and at zh ¼ 100%:
In order to capture the uncertainty involved in ground
seismicity, it is common to resort to MCS by considering
certain parameters of the problem as random variables [19].
The most rational choice here is the PGA, which is a
function of time t and of random parameter g: Since ground
accelerations eventually return to a quiescent state, there is
only a fluctuating component to contend with. A rather
standard choice for the autocovariance of the PGA is the
exponential correlation function, i.e.
covg ¼ k€xgðt;gÞ2l ¼ s2
g expð2Dt=tgÞf ðtÞ ð2Þ
In the above, k l denotes the expectation operator for the
ground accelerations €xgðtÞ; sg is their standard derivation, tg
is a correlation time interval, Dt is the time lapse and f ðtÞ is a
deterministic time envelope. It is well known that the
autocovariance and the PSDF form a Fourier transform pair
so once the latter has been determined, it is possible to
compute realizations of the PGA as
€xgðtÞ ¼ffiffi2
p XNj¼1
ffiffiffiffiffiffiffiffiffiffiffiffiSgðvjÞDv
qcosðvjt þ fÞf ðtÞ ð3Þ
Fig. 3. Seismic load: spatial variation of the load (a) across the horizontal
plane, (b) along the height and (c) spectrum-consistent synthetic ground
acceleration history.
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448438
where Dv ¼ ðv2 2 v1Þ=N is the frequency increment and fj
is a random phase angle uniformly distributed in the interval
[0,2p]. The frequency contents of the ground signal
falls between cut-off frequencies v1 ¼ 0:4p and v2 ¼ 50
p (rad/s), while the envelope function shown below is
essential in ameliorating the stationary assumption inherent
in this simulation procedure:
f ðtÞ ¼
ðt=T1Þ2; t # T1
1:0; T1 , t , T2
exp{ 2 cðt 2 T2Þ=TD}; t $ T2
8>><>>:
ð4Þ
Rise and drop times, respectively, range as
T1 ¼ 1:0–3:0 s and T2 ¼ 6:0–8:0 s, while total signal
duration is TD ¼ 15:0 s and c ¼ 5 is a dimensionless
constant. Obviously, as the number of samples N in Eq. (3)
increases, the realizations become a more realistic
representation of the original random field. Finally, Fig. 3
shows the spatial and temporal distribution of the input to
the tank from the ground motions, which are described by
synthetic acceleration signals. As with the chemical
explosion case, the Latin hypercube method will be used
to produce a family of statistically equivalent acceleration
records at a given PGA level, with the KT filter parameters
and the envelope function characteristic times assumed to
be random variables with uniform distributions. A total of
six PGA levels that produce the full range of damage,
from minor to severe, will be considered.
3. Methods of structural analysis
We now present details regarding the procedure used for
analysing the steel storage tank under the extreme transient
loads previously described.
3.1. Structural models
The specific tank examined herein [27] has clear height
h ¼ 29:5 m, mean base radius r ¼ 20:0 m and its thickness t
is a variable ranging from tb ¼ 21:9 mm at the base to
tt ¼ 9:52 mm at the top, as was previously shown in Fig. 1.
There is a stiffener beam around the tank’s perimeter at the
top, while the tank’s base is welded to a special metal alloy
ring plate, which in turn rests on a concrete foundation.
The tank material is ST 37 steel, with an elasticity modulus
E ¼ 2:1 £ 108 kN/m2, a Poisson’s ratio n ¼ 0:30 and a
mass density r ¼ 7850 kg/m3. The plastic modulus is
Epl ¼ E £ 1024; while the material yield stress is
sy ¼ 2:35 £ 105 kN/m2, and the yield strain and strain to
fracture are 1y ¼ 1:12% and 1ul ¼ 10:0%; respectively.
Additional, secondary design details that are particular to
such tanks are ignored.
The essential component in a FEM representation of
thin-walled, circular cylindrical tanks is development of
a simple, yet robust shell finite element that can be
incorporated into an open-end computer program capable
of transient non-linear analysis [20]. For this purpose,
we employed an in-house finite element program with pre-
and post-processing capabilities for data input/output built
around the facilities of the commercial program Nastran
[28]. Furthermore, the program employs Newmark’s
beta time integration algorithm, supplemented by the
Newton–Raphson iterative scheme for capturing non-linear
response with the tangent modulus concept.
Following standard convergence studies, the tank was
finally modelled by 288 quadrilateral shell finite elements
arranged in nine circumferential bands so as to account for
variable wall thickness and the presence of the stiffener
beam. Furthermore, this FEM model is detailed enough to
capture bending distortions near the lower edge of the tank’s
wall. This particular discretization required 320 nodal points
and the resulting mesh is shown in Fig. 1(c), where the
entire tank is modelled. Among the large volume of results
generated by the free-vibration analysis, we show
the dominant (first) mode in Fig. 4, namely combined
flexural – distortional mode at an eigenfrequency
f ¼ 2:21 Hz. When modal analysis was specified for the
two types of loadings previously described, which
are symmetric about the tank’s reference diameter and
piece-wise uniform across the height, the resulting
participation factor for this mode was very low (less than
g ¼ 0:01). Therefore, the mode with the largest
participation factor had to be identified, which turned out
to be the skew-symmetric, bending mode at f ¼ 19:99 Hz
given in Fig. 5. The two participation factors for the blast
load, one for the restoring forces (‘static’) and another for
the inertia forces (‘dynamic’), were gst ¼ 0:773 and
gdyn ¼ 0:486; respectively. As far as the seismic load was
concerned, the aforementioned participation factors
assumed values of gst ¼ 0:926 and gdyn ¼ 0:713;
respectively. It is interesting to note that both blast and
seismic loads, despite being two very different types of
events, share a similar spatial variation when applied to
this particular type of industrial structure and activate the
same mode.
Based on the results of the FEM eigenvalue analysis, a
continuous, cantilevered beam model (Fig. 1(d)) was
developed for the steel tank. Firstly, the bending stiffness
coefficients were computed by assuming a step-wise
variable thickness with unit value at the bottom and by
retaining the same thickness ratios of the original structure,
which had nine circumferential bands of constant thickness
decreasing from bottom to top. The height of the beam is the
same as that of the original tank, i.e. h ¼ 29:5 m.
Next, masses were lumped at the nine levels that demark
change in thickness, and full fixity was assumed at the base.
The value of the mass coefficient at the top node was
adjusted so as to produce the fundamental frequency of
f ¼ 19:99 Hz associated with the dominant (in terms of its
participation factors) mode and the remaining masses were
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448 439
adjusted by using the actual thickness ratios. Finally, a
nominal amount of modal damping of 4.5% was prescribed,
which is representative of rather flexible steel structures.
Thus, a nine-node model was established with two DOF per
node (translational and rotational) for a total of 18 active
DOF, which is far more efficient for the ensuing stochastic
analysis than the 3D FEM model.
As far as the loading sequence on this particular MDOF
model is concerned, we use a uniformly distributed
horizontal load for the blast, whose equivalent value peq is
computed by multiplying the original reference value P0 by
the inverse of the ratio defined as total tank mass divided by
the sum of lumped masses of the MDOF model. This ratio is
about 30:1, while the time variation of the equivalent load
remains unchanged. For the seismic event, we use a
step-wise uniform load whose value at the top is
peq ¼ 2mðlevel¼9Þ €xg; while the variation of peq with height
follows the thickness ratio of the original tank.
The formulation is now in terms of relative displacements,
while the time variation of the synthetic ground
accelerations remains unchanged.
The last task that remains is to adjust the material
parameters of the MDOF model for the purposes of a non-
linear, time-stepping analysis. For this purpose, a quasi-
static (‘push-over’) analysis of the 3D FEM model for a
step-wise uniformly distributed load across the height of the
tank is performed. The constitutive law used derives from
the bilinear, strain-hardening model for ductile steel,
whose values were prescribed earlier on. At each load
increment, the total base shear Vb in the FEM model is
computed, normalized by the tank’s total mass M; and
plotted in Fig. 6 versus maximum lateral deflection ut at the
tank’s top. The process is repeated until complete yielding
of the base is reached. By curve-fitting a bilinear plot to the
l ¼ Vb=M versus ut curve, a yield displacement
uy ¼ 0:0492 m can be identified. Next, the ‘pseudo-accel-
eration’ ly ¼ 559:8 m/s2 value that corresponds to uy is
converted into a load through multiplication by the tank
mass at each level, scaled by the total tank to MDOF model
mass ratio, and applied to the beam as a distributed load
(Fig. 6(c)). The moment computed at the base of the
continuous beam is the plastic (or yield) moment Mp:
Finally, this procedure was assumed to cover both blast and
seismic load cases.
Thus, considerable economy in scale has been reached,
given that a total of 18 DOF (later increased to 20 by
introducing base rotation and translation for SSI purposes)
are now sufficient to capture the basic mechanical behaviour
of the original tank. We note in passing that further
reduction in tank modelling was achieved in earlier work
[24,25] by introducing a SDOF oscillator, whose flexibility
was adjusted through energy balance considerations, i.e. by
specifying equal kinetic energies with the continuous beam
model. This was deemed necessary, since the rather
extensive MCS used within the context of risk analysis
required the simplest structural model possible for
efficiency purposes. Use of the Latin hypercube method
[17] in the present work, which generates a much smaller
sample space for comparable accuracy, allows use of the
present MDOF model.
3.2. Equations of motion
The MDOF model exhibiting elasto-plastic material
behaviour that was calibrated above will now be incorpor-
ated within the context of an efficient probabilistic analysis
for risk assessment. The basic parameters of this model are
mass matrix M; elastic tangent stiffness matrix K; and
damping matrix C built by specifying a damping coefficient
z ¼ 4:5% in each mode. In addition, it is necessary to
introduce ductility m as the ratio of maximum transient
response to yield displacement uy: The dynamic equilibrium
Fig. 4. FEM modal analysis of the steel tank: dominant first mode at
f ¼ 2:21 Hz with nearly zero participation factors for the two types of
transient loads.
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448440
equations of the MDOF structural system read as
M€yðtÞ þ C_yðtÞ þ RðtÞ ¼ FðtÞ ð5Þ
with yðtÞ denoting the vector of horizontal displacement and
rotational DOF and R representing the restoring force
vector. Specifically, in the elastic range, R ¼ KyðtÞ; while
beyond it, the system is plastic with much reduced stiffness
(the new coefficients are four orders of magnitude less than
the elastic ones). Lastly, vector FðtÞ corresponds to external
loads imposed on the structure. These are given by either
an equivalent, uniformly distributed load PðtÞ along the
height with ‘triangular’ time variation for chemical
explosions or by an inertia term 2MI€xgðtÞ applied along
the height of the structure for earthquake motions
(where IT ¼ ½1; 0; 1;…; 0�). In the latter case, the horizontal
displacements uiðtÞ are defined as relative to the ground
motion xgðtÞ: We note here that both loading cases are
viewed as random processes, so as to allow for statistical
generation of families of load records.
Eq. (5) constitute an initial value problem. The resulting
system of non-linear, second-order differential equations are
numerically integrated by the generalized energy
momentum algorithm [29], which in turn is based on the
generalized alpha method [30] for linear systems. This new
algorithm adopts both linear and angular momentum
conservation criteria during the current time-step to avoid
Fig. 5. FEM modal analysis of the steel tank: non-symmetric bending mode at f ¼ 19:99 Hz with the highest participation factor for the two types of transient
loads.
Fig. 6. Nonlinear FEM analysis of the steel tank: (a) monotonically
increasing lateral piece-wise uniform load, (b) computation of the plastic
moment for the continuous beam model and (c) normalized base shear
versus top horizontal deflection curves.
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448 441
loss of unconditional stability that is commonly associated
with elastic systems, along with desirable numerical
dissipation characteristics. As an illustration of the type
of results that were recovered, Fig. 7 shows the evolution of
crown displacement u9ðtÞ and base moment MbðtÞ for the
empty steel tank with total mass M ¼ 403:34 t
(that corresponds to an equivalent MDOF model mass of
meq ¼ 12:324 t), which is subjected to a ‘reference’
explosive load (corresponding to load level ‘4’ in a scale
from 1–6) of magnitude P0 ¼ 246; 453 kN with linear time
decay and time duration td ¼ 0:1453 s, which exceeds the
mean duration of the event by a factor of 1.60. Also, Fig. 8
plots the same quantities, but for the synthetic earthquake
acceleration record with ‘reference’ PGA of €xg ¼ 0:38 g
and total duration TD ¼ 15:0 s, which also corresponds to a
load level of ‘4’. In both cases, 750 time steps were used in
the time integration algorithm and the criterion for yielding
was MbðtÞ $ Mp: Note that plots for SSI effects, which
will be discussed in Section 3.3, are also included in
these figures. It is interesting to note that a typical
computer run for the MDOF model consumed a CPU time
of less than 1.0 s.
Although the MDOF model cannot provide all the details
that a 3D FEM model does, it gives a good overall picture of
the dynamic response, which is on the conservative side.
Also, it is well suited for intense sudden loads that produce
non-linear response, since in this case mostly lower modes
of vibration contribute to the gross displacement picture. If
additional effects such as instability need to be studied, then
recourse must be made to the FEM model. In principle,
risk analysis is still possible, but becomes extremely
time-consuming because a very large number of fully
Fig. 7. Time evolution of the MDOF steel tank response for a chemical blast yielding horizontal load P0 ¼ 246; 453 kN : (a) top horizontal displacement u9 and
(b) base moment Mb:
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448442
non-linear time-stepping analyses need to be performed at
each load level.
3.3. Soil–structure interaction effects
SSI effects are manifested because of ground flexibility
and are important for the purposes of this work, given the
large mass (when the tank is full) and rather heavy
foundation design of most cylindrical steel storage tanks.
Since we seek the simplest model possible, the influence of
the explosion on the ground itself is neglected in the
chemical blast case, while kinematic interaction effects that
have to do with the influence of the foundation shape on the
earthquake signal that develops at the base of structure
were not taken into account. Thus, we use a set of
frequency-independent soil impedances, dampers and
masses [21], which for the case of a circular rigid foundation
of radius r resting on the ground surface are:
Kh ¼ 8Gr=ð2 2 nÞ; Ch ¼ 1:08
ffiffiffiffiffiffiffiffiKhrr3
q;
Mh ¼ 0:28rr3
ð6Þ
Ku ¼ 8Gr3=ð3ð1 2 nÞÞ; Cu ¼ 0:47
ffiffiffiffiffiffiffiffiKurr3
q;
Iu ¼ 0:49rr5
In the above formulas, subscripts h, u; respectively, denote
horizontal and rotational DOF, as shown in Fig. 1(e). We
assume relatively soft soil [21] with the following numerical
values for its properties: Poisson’s ratio n ¼ 0:25; shear
modulus G ¼ 72:0 £ 106 kN/m2, density r ¼ 1800 kg/m3
and shear wave-speed cs ¼ffiffiffiffiffiffiffiðG=rÞ
p¼ 200 m/s. Basically,
the influence of soil on the response of the overlying structure
Fig. 8. Time evolution of the MDOF steel tank response for seismic motions with a peak ground acceleration €xg ¼ 0:38g : (a) top horizontal displacement u9
and (b) base moment Mb:
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448 443
is elongation of the fundamental period of vibration, an effect
that becomes more pronounced as the ratio of stiffness of
structure to that of soil increases. SSI is not necessarily
detrimental and, in fact, may lead to stress reduction in the
structure at the expense of a more pronounced kinematic
state. For stochastic loads, it usually leads an increase in the
probability for a certain type of damage to occur at a given
load level [24,25].
Thus, the additional equations of dynamic equilibrium
for the horizontal displacement u0 and angle of rotation u0 at
the foundation (level 0) are added to Eq. (5) as follows:
The above is now a 20 DOF system, where Iu is the mass
moment of inertia corresponding to lumped mass Mh, and in
both augmented damping and stiffness matrices various
coupling terms (with mixed indices) appear. Numerical
results from computations involving the MDOF system with
SSI are also plotted in Figs. 7 and 8 for the blast and
earthquake loads, respectively. Two types of soil are
considered whose lateral stiffness varies by two orders of
magnitude, namely the soft soil given above
(with Kh ¼ 2:1 £ 109 kN/m) and a stiff soil (with
Kh ¼ 210 £ 109 kN/m). Comparisons for the blast load
reveal an increase (of roughly 20%) in the tank’s kinematic
response as time evolves. At the same time, this phenom-
enon is accompanied by a slight increase (of about 2–5%) in
the base moment. As far as the seismic loads go, the
presence (or absence) of SSI is of minor importance since all
three curves (fixed, soft soil, stiff soil) lie close together
through-out the time duration of the ground accelerations.
4. Risk analysis
The risk analysis procedure is broken down into the
following two basic steps:
4.1. Generation of structure and load samples
Risk is broadly defined as the convolution of hazard with
consequence, i.e. the probability of occurrence of an event
that can lead to structural (or other) type of failure times the
economic (or other) loss associated with this event. Risk
reduction is essentially a stochastic optimization problem
[1], and additional aspects that should be considered is the
influence of various structural components (sensitivity
analysis), along with changes that can be implemented so
as to reduce risk. On the hazard side of the equation, it is
necessary to reduce existing failure probability, which is
directly dependent on the randomness exhibited in the
resistance of the structure as well as in the loading. In
general, when performing reliability analyses for non-linear
systems, the capacity as well as the demand in conventional
structures is assumed to be log-normally distributed [31,32].
4.1.1. Blast load case
In order to model blast-induced loading for risk analysis,
we define a sequence of blasts with increasing strength
levels. Then, we integrate pressure along the entire circular
cylindrical tank configuration to produce a net, uniformly
distributed horizontal load with a simplified triangular time
variation PðtÞ ¼ P0ð1 2 t=tdÞ; where P0 is the value
registered at each load level and td is its time duration, as
was discussed in Section 2.1. For a cylindrical steel tank
with overall dimensions h ¼ 29:5 m, r ¼ 21:0 m, the
maximum load resulting from an intermediate blast giving
80.0 kPa of overpressure (load level ‘4’ in a scale from 1 to
6) is p ¼ 8354 kN/m, based on a doubling caused by the
reflected shock. The maximum equivalent horizontal
uniform load is therefore P ¼ 369; 680 kN at that level,
but in order to be consistent with negative pressure and other
effects that are produced as the blast engulfs the entire tank,
the total impulsive load is two-thirds of this value, i.e.
P0 ¼ 246; 453 kN. Finally, mean time duration of the blast
td is given by ratio 2r=U ¼ 0:0908 s; where U is the pressure
wave-speed from the Rankine–Hugoniot relations.
4.1.2. Structural configuration
As far as the structural configuration is concerned, we
consider certain key parameters to be random variables.
These are the yield (or plastic) moment Mp whose mean
value is 6300 kN m and its standard deviation is 200 kN m,
the damping coefficient z whose range is 3–6%, and the
fluid mass Mfl stored in the tank. We assume the tank stores
hydrocarbon fluids with mass density rfl ¼ 0:5 kg=m3; and
the mean value corresponds to a half-empty configuration
that gives Mfl ¼ 9227 ton (the equivalent MDOF
model fluid mass is mfl ¼ 300 ton). This fluid mass is
lumped at the MDOF model node that is at a distance h=3
from the bottom of the tank. By prescribing a standard
deviation of 50t; the samples that will be generated cover
the range from nearly empty to nearly full. Also, the bilinear
elastoplastic material model uses an elastic bending stiffness
of EI and a corresponding plastic stiffness EIpl ¼ 1024EI:
By specifying the above range and assuming a normal
distribution for these key structural parameters, the Latin
hypercube method commences generation of a family of
storage tanks. These tank samples are then matched with a
family of blast records, which come from perturbing the
time duration td of the horizontal load. This last task is done
M 0
0Mh 0
0 Iu
26664
37775
€y
€u0
€u0
8>>><>>>:
9>>>=>>>;þ
C Cij
Cji
Chh Chu
Cuh Cuu
26664
37775
_y
_u0
_u0
8>>><>>>:
9>>>=>>>;
þ
K Kij
Kji
Khh Khu
Kuh Kuu
26664
37775
y
u0
u0
8>>><>>>:
9>>>=>>>;¼
FðtÞ
F0
0
8>>><>>>:
9>>>=>>>;
ð7Þ
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448444
by considering the range of 0:10td –2:0td; which like all load
parameters follows a uniform distribution. At this point, the
user may specify a minimum number of structure–load
pairs (or samples), which in our case was taken as N ¼ 100;
following parametric studies in which the same results were
recovered for larger values of N: Finally, we specify six load
levels that range from 0:25P0 to 1:75P0; computed so as to
produce structural behaviour starting from mild elastic
response up to pronounced non-linearities. Thus, at each
load level, the MDOF model is solved at least N times and
an equal number of ductility values m are obtained. These
values are statistically processed to yield mean value mL and
second-order reliability index bL (Table 1).
Next, five limit states are defined for the structure, as
listed in Table 2. To each such state, the corresponding
ductility capacity (or resistance factor R) is assumed to obey
a logarithmic distribution with mean value mR and second
moment reliability index bR: It should be noted here that the
second-order reliability index is equal to the logarithmic
standard deviation of the structural capacity population
(the same holds true for the structural response as well). It is
subsequently used, along with the relevant mean value, to
evaluate limit state probabilities. Finally, this procedure was
repeated for the MDOF model with SSI so as to capture the
influence of ground flexibility on the ductility. The relevant
results appear in Tables 3 and 4 for the stiff and soft soils,
respectively. We mention in passing that the ground
parameters were not perturbed, which implies a fixed
foundation configuration.
4.1.3. Seismic load case
Next, we focus on seismic loads, which are characterized
by the PGA output described in Section 2.2. At a fixed
acceleration level, we again discern a number of possible
variations prior to activating the realizations given by
Eqs. (1)–(4) for producing synthetic accelerograms. These
are produced by assuming the KT filter frequency vk and
damping zk; as well as time the envelope rise time T1; are all
random variables with uniform distributions whose range is
3p–9p rad/s, 0.15–0.25% and 1.5–3.0 s, respectively.
We furthermore define six levels for the peak white noise
intensity S0 of the earthquake signal ranging from 0.005 to
0.0394 m2/s3, in equal increments. These correspond to PGA
levels of 0:15g–0:45g: Thus, at each PGA value, the
synthetic ground motion family is matched with
the structural model family of samples within the context
of the Latin hypercube method and at least N ¼ 100 pairs
are solved, yielding an equal number of ductility values m:
These values were again statistically processed, assuming
a logarithmic distribution, to give a mean value mL and
a second moment reliability index bL; as shown in Table 5.
Since soil flexibility was of little consequence to the tank’s
response to ground accelerations, no additional compu-
tations for SSI effects were performed here.
4.2. Fragility curves
The last step is to introduce the fragility of a structure
with respect to a given limit state, which is defined as
Table 1
Logarithmic distribution of structural response ðLÞ to blast loads in terms of
ductility (MDOF model)
Explosion load
Level
Mean value,
mL
Second moment
reliability index,
bL
1 0.2989 0.5261
2 0.5978 0.5261
3 1.1186 0.6799
4 2.3262 0.8924
5 4.3843 1.0394
6 7.3092 1.1396
Table 2
Logarithmic distribution of structural capacity ðRÞ in terms of ductility
Limit state Mean
value, mR
Second moment
reliability index, bR
Non-structural damage 1.0 0.5
Minor damage 2.0 0.5
Moderate damage 4.0 0.5
Severe damage 6.0 0.5
Collapse 7.5 0.5
Table 3
Logarithmic distribution of structural response ðLÞ to blast loads in terms of
ductility (MDOF model with SSI effects for Ku ¼ 210 £ 109 kN/m)
Explosion load
level
Mean value, mL Second moment
reliability index,
bL
1 0.2588 0.5350
2 0.5177 0.5350
3 0.8217 0.5838
4 1.5972 0.7884
5 2.9335 0.9458
6 4.8959 1.0621
Table 4
Logarithmic distribution of structural response to blast loads ðLÞ in terms of
ductility (MDOF model with SSI effects for Ku ¼ 2:1 £ 109 kN/m)
Explosion load
level
Mean value, mL Second moment
eliability index,
bL
1 0.2546 0.5883
2 0.5091 0.5883
3 0.8325 0.6572
4 1.7324 0.8917
5 3.2938 1.0628
6 5.6269 1.1817
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448 445
the probability Pf of load L exceeding resistance R and is
given by
Pf ¼ PðR # LÞ ¼ð1
0½1 2 FsðrÞ�fRðrÞdr ð8Þ
where Fs is the cumulative probability distribution of L and
fR is the probability density function of R: For both R and L
log-normally distributed, which is common assumption
[32], Eq. (8) can be written as
Pf ¼ Fð2‘nðmR=mLÞ=
ffiffiffiffiffiffiffiffiffiffiffis2
R þ s2L
qÞ ð9Þ
where F is the standardized normal distribution function.
All results from the previous sub-section can now be
summarized in the fragility curves of Figs. 9 and 10, which
plot limit state damage probability Pf versus blast-induced
net horizontal force and normalized PGA, respectively. In
the former case, the first graph is for the standard fixed-base
MDOF model, while the second and third graphs are for SSI
with stiff and soft soils, respectively. In the later case, there
is only one graph for the fixed-base MDOF model. In
parallel, Tables 6–9 give all Pf values that were used in
constructing the aforementioned graphs. We observe that
for blast loads, the failure probability is more pronounced as
compared to failure probability for seismic motions, at least
for the range of loads that were considered here. Apparently,
the impulsive character of the blast loads falling on what is
essentially a flexible structure, especially in the presence of
the extra fluid mass it stores, is more detrimental than the
relatively slowly developing phenomenon of ground
motions. Finally, in order to look at the effect of SSI more
closely for blast loads, we compare Pf for two values of the
ground horizontal stiffness that are two orders of magnitude
apart (Kh ¼ 210 £ 109 and 2.1 £ 109 kN/m). As the soil
progressively softens, damage probability clearly increases
(with the Pf curves moving ‘upwards’), reflected by the fact
that the overall structural system has more inherent
uncertainty. It should be noted here that Pf ¼ 1:0 denotes
certain failure. Thus, it is possible for blast loads of a
fixed level of intensity to produce more damage in the
presence of foundation flexibility, as compared to the
absence of such an effect.
Based on these results, the following general conclusions
can be drawn: (i) for every damage level, the probability that
a certain structural configuration will exhibit the type of
damage associated with it, increases with increasing load
level; (ii) for every load level, the probability that a structure
will exhibit damage greater than a certain predefined
Table 5
Logarithmic distribution of structural response ðLÞ to seismic loads in terms
of ductility (MDOF model)
Seismic load level Mean value mL Second moment
reliability index bL
1 0.1989 0.3680
2 0.2956 0.3460
3 0.3703 0.3847
4 0.4166 0.3506
5 0.4958 0.3704
6 0.5384 0.3558
Fig. 10. Fragility curves for a steel storage tank under seismically induced
ground motions without SSI effects.
Fig. 9. Fragility curves for a steel storage tank under chemical blast loads:
(a) without SSI effects, (b) with SSI effects for stiff soil ðKh ¼ 210 £ 109
kN=mÞ and (c) with SSI effects for soft soil ðKh ¼ 2:1 £ 109 kN=mÞ:
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448446
Table 7
Fragility matrix for atmospheric blast loads (MDOF model with SSI effects for Ku ¼ 210 £ 109 kN=m)
Load level (kN) Level of damage
Non-structural
damage
Minor
damage
Moderate
damage
Severe
damage
Collapse
61613.3 0.03247 0.00262 0.00009 0.00001 0.00000
123226.6 0.18430 0.03247 0.00262 0.00041 0.00013
184839.9 0.39919 0.12359 0.01975 0.00485 0.00201
246453.2 0.69201 0.40482 0.16272 0.07815 0.04880
308066.5 0.84278 0.63984 0.38596 0.25179 0.19013
369679.8 0.91198 0.77715 0.56835 0.43124 0.35819
Table 6
Fragility matrix for atmospheric blast loads (MDOF model)
Load level (kN) Level of damage
Non-structural
damage
Minor
damage
Moderate
damage
Severe
damage
Collapse
61613.3 0.04808 0.00441 0.00018 0.00002 0.00000
123226.6 0.23924 0.04808 0.00441 0.00074 0.00025
184839.9 0.55282 0.24555 0.06553 0.02328 0.01207
246453.2 0.79540 0.55871 0.29809 0.17715 0.12623
308066.5 0.89999 0.75191 0.53170 0.39281 0.32079
369679.8 0.94502 0.85115 0.68595 0.56301 0.49174
Table 8
Fragility matrix for atmospheric blast loads (MDOF model with SSI effects for Ku ¼ 2:1 £ 109 kN=m)
Load level (kN) Level of damage
Non-structural
damage
Minor
damage
Moderate
damage
Severe
damage
Collapse
61613.3 0.03818 0.00379 0.00018 0.00002 0.00001
123226.6 0.19095 0.03818 0.00379 0.00070 0.00025
184839.9 0.41217 0.14427 0.02867 0.00838 0.00388
246453.2 0.70455 0.44413 0.20652 0.11215 0.07587
308066.5 0.84493 0.66450 0.43432 0.30481 0.24177
369679.8 0.91090 0.78992 0.60486 0.48005 0.41140
Table 9
Fragility matrix for seismic loads (MDOF model)
Load level (%g) Level of damage
Non-structural
damage
Minor damage Moderate damage Severe damage Collapse
0.153 0.00464 0.00010 0.00000 0.00000 0.00000
0.222 0.02251 0.00083 0.00001 0.00000 0.00000
0.281 0.05767 0.00375 0.00008 0.00001 0.00000
0.325 0.07580 0.00510 0.00011 0.00001 0.00000
0.379 0.12980 0.01250 0.00040 0.00003 0.00001
0.418 0.15650 0.01624 0.00054 0.00004 0.00002
D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448 447
level decreases as the level of damage increases; (iii) due to
the additional effect of foundation flexibility, the probability
for a certain damage level to occur at fixed load intensity
shows an increase with decreasing ground stiffness; and (iv)
blast loads are more critical for a tank, more so as the mass
of the fluid it stores increases, while seismic loads seem to
have a relatively minor effect on such a flexible structure,
especially when the mass of the stored fluid is low.
5. Conclusions
An integrated numerical approach has been developed for
assessing risk posed by chemical explosions and earthquake
motions to typical industrial structural facilities. The analysis
yields, as a final product, a set of fragility curves that allow
computation of failure probabilities for steel storage tanks
due to overpressure caused by a chemical explosion or due to
ground acceleration caused by an earthquake. These curves
are useful within the context of planning prevention and
intervention strategies for large industrial complexes, where
the type and intensity of the external loads are key parameters
in estimating the level of damage sustained by various
structural units. In addition, the presence of compliant soil
causes SSI phenomena that increase the probability for a
certain damage level to occur at fixed load intensity. In
closing, it should be noted that the present methodology is
generally enough to be expanded to other categories of
structures (e.g. buildings, bridges) as well as to other types of
extreme environmental and man-induced loads (e.g. high
winds, ocean waves, fire hazards).
Acknowledgements
The authors wish to thank the Greek General Secretariat
for Research and Technology for its financial support of the
project No. 1573 entitled ‘Development of Methodologies
for Evaluating Risk in Industrial Units due to Explosions’.
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